We present a novel Bayesian approach to the problem of value function estimation in continuous state spaces. We define a probabilistic generative model for the value function by imposing a Gaussian prior over value functions and assuming a Gaussian noise model. Due to the Gaussian nature of the random processes involved, the posterior distribution of the value function is also Gaussian and is therefore described entirely by its mean and covariance. We derive exact expressions for the posterior process moments, and utilizing an efficient sequential sparsification method, we describe an on-line algorithm for learning them. We demonstrate the operation of the algorithm on a 2-dimensional continuous spatial navigation domain.

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